35 algorithm-types
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35 algorithm-types

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Study materrial for understanding algorithms and cryptography.

Study materrial for understanding algorithms and cryptography.

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35 algorithm-types 35 algorithm-types Presentation Transcript

  • Types of Algorithms
  • Algorithm classification
    • Algorithms that use a similar problem-solving approach can be grouped together
    • This classification scheme is neither exhaustive nor disjoint
    • The purpose is not to be able to classify an algorithm as one type or another, but to highlight the various ways in which a problem can be attacked
  • A short list of categories
    • Algorithm types we will consider include:
      • Simple recursive algorithms
      • Backtracking algorithms
      • Divide and conquer algorithms
      • Dynamic programming algorithms
      • Greedy algorithms
      • Branch and bound algorithms
      • Brute force algorithms
      • Randomized algorithms
  • Simple recursive algorithms I
    • A simple recursive algorithm :
      • Solves the base cases directly
      • Recurs with a simpler subproblem
      • Does some extra work to convert the solution to the simpler subproblem into a solution to the given problem
    • I call these “simple” because several of the other algorithm types are inherently recursive
  • Example recursive algorithms
    • To count the number of elements in a list:
      • If the list is empty, return zero; otherwise,
      • Step past the first element, and count the remaining elements in the list
      • Add one to the result
    • To test if a value occurs in a list:
      • If the list is empty, return false; otherwise,
      • If the first thing in the list is the given value, return true; otherwise
      • Step past the first element, and test whether the value occurs in the remainder of the list
  • Backtracking algorithms
    • Backtracking algorithms are based on a depth-first recursive search
    • A backtracking algorithm:
      • Tests to see if a solution has been found, and if so, returns it; otherwise
      • For each choice that can be made at this point,
        • Make that choice
        • Recur
        • If the recursion returns a solution, return it
      • If no choices remain, return failure
  • Example backtracking algorithm
    • To color a map with no more than four colors:
      • color(Country n)
        • If all countries have been colored (n > number of countries) return success; otherwise,
        • For each color c of four colors,
          • If country n is not adjacent to a country that has been colored c
            • Color country n with color c
            • recursivly color country n+1
            • If successful, return success
        • Return failure (if loop exits)
  • Divide and Conquer
    • A divide and conquer algorithm consists of two parts:
      • Divide the problem into smaller subproblems of the same type, and solve these subproblems recursively
      • Combine the solutions to the subproblems into a solution to the original problem
    • Traditionally, an algorithm is only called divide and conquer if it contains two or more recursive calls
  • Examples
    • Quicksort:
      • Partition the array into two parts, and quicksort each of the parts
      • No additional work is required to combine the two sorted parts
    • Mergesort:
      • Cut the array in half, and mergesort each half
      • Combine the two sorted arrays into a single sorted array by merging them
  • Binary tree lookup
    • Here’s how to look up something in a sorted binary tree:
      • Compare the key to the value in the root
        • If the two values are equal, report success
        • If the key is less, search the left subtree
        • If the key is greater, search the right subtree
    • This is not a divide and conquer algorithm because, although there are two recursive calls, only one is used at each level of the recursion
  • Fibonacci numbers
    • To find the n th Fibonacci number:
      • If n is zero or one, return one; otherwise,
      • Compute fibonacci(n-1) and fibonacci(n-2)
      • Return the sum of these two numbers
    • This is an expensive algorithm
      • It requires O(fibonacci(n)) time
      • This is equivalent to exponential time, that is, O(2 n )
  • Dynamic programming algorithms
    • A dynamic programming algorithm remembers past results and uses them to find new results
    • Dynamic programming is generally used for optimization problems
      • Multiple solutions exist, need to find the “best” one
      • Requires “optimal substructure” and “overlapping subproblems”
        • Optimal substructure : Optimal solution contains optimal solutions to subproblems
        • Overlapping subproblems : Solutions to subproblems can be stored and reused in a bottom-up fashion
    • This differs from Divide and Conquer, where subproblems generally need not overlap
  • Fibonacci numbers again
    • To find the n th Fibonacci number:
      • If n is zero or one, return one; otherwise,
      • Compute, or look up in a table, fibonacci(n-1) and fibonacci(n-2)
      • Find the sum of these two numbers
      • Store the result in a table and return it
    • Since finding the n th Fibonacci number involves finding all smaller Fibonacci numbers, the second recursive call has little work to do
    • The table may be preserved and used again later
  • Greedy algorithms
    • An optimization problem is one in which you want to find, not just a solution, but the best solution
    • A “greedy algorithm” sometimes works well for optimization problems
    • A greedy algorithm works in phases: At each phase:
      • You take the best you can get right now, without regard for future consequences
      • You hope that by choosing a local optimum at each step, you will end up at a global optimum
  • Example: Counting money
    • Suppose you want to count out a certain amount of money, using the fewest possible bills and coins
    • A greedy algorithm would do this would be: At each step, take the largest possible bill or coin that does not overshoot
      • Example: To make $6.39, you can choose:
        • a $5 bill
        • a $1 bill, to make $6
        • a 25¢ coin, to make $6.25
        • A 10¢ coin, to make $6.35
        • four 1¢ coins, to make $6.39
    • For US money, the greedy algorithm always gives the optimum solution
  • A failure of the greedy algorithm
    • In some (fictional) monetary system, “krons” come in 1 kron, 7 kron, and 10 kron coins
    • Using a greedy algorithm to count out 15 krons, you would get
      • A 10 kron piece
      • Five 1 kron pieces, for a total of 15 krons
      • This requires six coins
    • A better solution would be to use two 7 kron pieces and one 1 kron piece
      • This only requires three coins
    • The greedy algorithm results in a solution, but not in an optimal solution
  • Branch and bound algorithms
    • Branch and bound algorithms are generally used for optimization problems
      • As the algorithm progresses, a tree of subproblems is formed
      • The original problem is considered the “root problem”
      • A method is used to construct an upper and lower bound for a given problem
      • At each node, apply the bounding methods
        • If the bounds match, it is deemed a feasible solution to that particular subproblem
        • If bounds do not match, partition the problem represented by that node, and make the two subproblems into children nodes
      • Continue, using the best known feasible solution to trim sections of the tree, until all nodes have been solved or trimmed
  • Example branch and bound algorithm
    • Travelling salesman problem: A salesman has to visit each of n cities (at least) once each, and wants to minimize total distance travelled
      • Consider the root problem to be the problem of finding the shortest route through a set of cities visiting each city once
      • Split the node into two child problems:
        • Shortest route visiting city A first
        • Shortest route not visiting city A first
      • Continue subdividing similarly as the tree grows
  • Brute force algorithm
    • A brute force algorithm simply tries all possibilities until a satisfactory solution is found
      • Such an algorithm can be:
        • Optimizing : Find the best solution. This may require finding all solutions, or if a value for the best solution is known, it may stop when any best solution is found
          • Example: Finding the best path for a travelling salesman
        • Satisficing : Stop as soon as a solution is found that is good enough
          • Example: Finding a travelling salesman path that is within 10% of optimal
  • Improving brute force algorithms
    • Often, brute force algorithms require exponential time
    • Various heuristics and optimizations can be used
      • Heuristic : A “rule of thumb” that helps you decide which possibilities to look at first
      • Optimization : In this case, a way to eliminate certain possibilites without fully exploring them
  • Randomized algorithms
    • A randomized algorithm uses a random number at least once during the computation to make a decision
      • Example: In Quicksort, using a random number to choose a pivot
      • Example: Trying to factor a large prime by choosing random numbers as possible divisors
  • The End